In this paper, we introduce the \(N\times N\)  symmetric tridiagonal matrix-valued process \(H_{\alpha }(t)\)  defined by (REF ), associated with G\(\beta \) E. More precisely, \(H_{\alpha }(t)\)  is constructed by putting independent Brownian motions and Bessel processes with dimension \(\alpha _k,k=1,\dots ,N-1\)  on diagonal entries and off-diagonal ones, respectively. Note that \(H_{\alpha }(t)\)  does not meet the sufficient condition that their eigenvalue process satisfies (REF ) as reported in [1]}. In our main result (Theorem REF ), we derive the SDEs (REF ) of eigenvalue process of \(H_{\alpha }(t)\)  for general \(N >0\) , and we conclude that several eigenvalues of indexed principal minor sub-matrices of \(H_{\alpha }(t)\)  appear in the SDEs. Hence, the SDEs that eigenvalue processes of \(H_{\alpha }(t)\)  satisfy are non-Markov type. Formally, drift terms of the obtained SDEs tell us that the eigenvalues of \(H_{\alpha }(t)\)  interact with their minor eigenvalues, and these interactions are similar to that of Dyson's Brownian motion (REF ), see Remark REF . Moreover, we characterize the sufficient condition that eigenvalue processes of \(H_{\alpha }(t)\)  never collide with each other almost surely, by using the dimensions \(\alpha _k,k=1,\dots ,N-1\)  (and starting points) of Bessel processes. As a corollary, for \(2\times 2\)  (sub-)matrices, we recover the construction of eigenvalue processes done in [2]}. We remark some related topics for minor eigenvalue processes. In the each case of symmetric, hermitian and symplectic-matrix valued process, namely Dyson's Brownian motion with \(\beta =1,2\)  and 4 respectively, the eigenvalue process with their minor eigenvalues has been studied in [3]}. Adler, Nordenstam and van Moerbeke proved that eigenvalue processes of two consecutive minors for these matrices are diffusion processes, and they derive the SDEs that these processes satisfy. Our SDEs (REF ) slightly look like their SDEs in [3]}, however, ours are more complicated. They also showed that in the case of three consecutive minors, eigenvalue processes with minor eigenvalues are non-Markov. On the other hand, in the context of non-intersecting diffusion processes, multilevel versions of Dyson's Brownian motions are studied [5]} and known as corner process. Note that the eigenvalue processes with their minors and corner processes are very different processes as mentioned in [3]}.
